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Topological Transition in the Fermi Surface of Cuprate Superconductors in the Pseudogap Reg

a r X i v :c o n d -m a t /0607777v 4 [c o n d -m a t .s u p r -c o n ] 1 M a y 2007

Topological Transition in the Fermi Surface of Cuprate Superconductors in the

Pseudogap Regime

C.M.Varma and Lijun Zhu

Department of Physics and Astronomy,University of California,Riverside,California 92521,USA

The angle-resolved photoemission spectroscopy studies on cuprates in the pseudogap region reveal an extraordinary topological transition in which the ground state changes from one with a usual Fermi surface to one with four Fermi points.Such a state is not possible without some symmetry breaking which allows interference between one-particle basis states which is normally forbidden.We also show that the experimental results are quantitatively given without any free parameters by a theory and discuss the implications of the results.

PACS numbers:74.25.Jb,74.20.-z,74.72.-h

Recently,the single-particle spectral function A (k ,ω)obtained by angle-resolved photoemission spectroscopy (ARPES)on eight di?erent underdoped BSCCO cuprates with T c ranging from 25K to 90K has been systematically analyzed [1].The principal conclusion is

that the angular region in ?k

where A (k ,ω)has a max-ima at the chemical potential μis a universal function Φ(T/T g (x )).T is the temperature and T g (x )is the pseu-dogap transition temperature at a given doping level x ,the deviation of the density of holes from half-?lling.At T ≥T g (x ),Φ=2π,i.e.the angular region encloses an area,while for T →0,Φ→0and the region shrinks to 4points.T g (x )is,within the uncertainties of its de-termination,the same as that obtained from the resis-tivity or the thermodynamic measurements such as the magnetic susceptibility and the speci?c heat which also scale as functions of T/T g (x )[2].T g (x )is also consis-tent with the temperature at which time-reversal sym-metry (TRS)breaking is observed by dichroic ARPES experiments [3,4]in the same compounds.Qualitatively similar conclusions were arrived at by Yoshida et al.[5].Recall that in a Fermi liquid [6][or a marginal Fermi liquid (MFL)[7]],A (k ,ω)is maximum at ω=μfor k =k F ,thus de?ning the Fermi surface.The concept of a Fermi surface,properly de?ned as the nonanalytic surface which separates the area of occupied states from the unoccupied states in a Fermi liquid,is strictly mean-ingful only at T →0.This is merely a technicality in the usual metallic states.But for underdoped cuprates,the proper de?nition of a Fermi surface is essential.The experimental results suggest that for x >x c ,such that T g (x c )=0,the extrapolated T =0non-superconducting ground state has a Fermi surface,whereas for x

It has become the custom to call the angular region Φ(T/T g ),a Fermi arc .This is only harmless if the em-pirical procedure used to derive Φ(T/T g )from the exper-iments is kept in mind.

The truly new physics is the deduction that at T →0,a gap in the single-particle excitation spectra develops at the chemical potential for a range of electronic density at all angles on the Fermi surface except four nodal points.Of course,this deduction is based on extrapolating data

(available from about T =400K to 25K )down to T =0,but the data (reproduced in Fig.3below)is persuasive.It joins the list of novel concepts brought to physics by the cuprates because it does not obey Bloch’s counting theorem for periodic systems that gaps can only occur at Brillouin zone boundaries,i.e.,for integer ?llings [8,9].The theorem originally derived for non-interacting elec-trons also holds for interacting systems in which there is a one-to-one correspondence between particles and quasi-particles.But the derivation implies an even greater gen-erality,because it depends only on the (usually safe)as-sumption that the excitation energy of one-particle states is a single-valued continuous function of their momentum quantum number except at the zone-boundaries,where multiple energies are obtained for a given momentum due to interference.

The only previously known state [10],in the absence of disorder,where a gap (or a gap with nodal points or lines as in anisotropic superconductors)occurs tied to the chemical potential independent of band-?lling is su-perconductivity.In that case,the energy of states is a double valued function of momentum for states near k F with a gap.Bloch’s theorem is circumvented in this case only because the single-particle excitations near k =k F are linear combinations of electrons and holes and there-fore are not eigenstates of charge.A gap occurs due to an interference between basis states of di?erent charges.Circumventing Bloch’s counting theorem in other cir-cumstances requires that some other normally sacrosanct quantum number be no longer protected,which allows in-terference between basis states that is usually forbidden [11].

Bloch’s theorem is circumvented in a theory of the pseudogap state [14,15]for reasons not having to do with BCS pairing.This theory predicts that the pseudogap state breaks time-reversal symmetry below T g (x )with-out changing the translational symmetry.This aspect has been tested in BSCCO by dichroic ARPES [3,4]and in YBCO by polarized neutron scattering [16].The sta-tistical mechanical model derived to obtain such a sym-metry breaking [17]is the Ashkin-Teller model,which in the relevant range of parameters has a smooth change in speci?c heat at the transition with the entropy released

2 over a temperature range typically more than twice the

transition temperature[18].It is also shown that a nor-

mal Fermi surface cannot exist in such a TRS breaking

state.The order parameter is not a conserved quantity;

therefore,fermions with crystal momentum k have a?-

nite coupling g(k)to the order parameter?uctuations in

the long wavelength limit.Since in this limit the energy

of the order parameter?uctuations goes to0,a Fermi sur-

face instability with no change in translational symmetry

but in the harmonic of g(k)accompanies the broken TRS.

A stable state is found with an anisotropic gap in the

excitation spectra.In the new state,the single-particle

excitations are not eigenstates of crystal momentum but

formed from the linear combination of states of momen-

tum in a small region around a given momentum[14].

Interference between basis states of crystal momentum

leads to the anisotropic gap at the erstwhile Fermi sur-

face with four Fermi points left intact.The ground state

itself is perfectly periodic and therefore an eigenstate of

crystal momentum.This is not new conceptually;re-

call that in superconductivity the ground state conserves

charge although the single-particle excitations do not.

The purpose of this Letter is to show that the pre-

dictions of such a state agree quantitatively with the

new experimental results.We calculate the function

Φ(T/T g(x)),de?ned above and compare it with the ex-

periments[1].An independent experimental result con-

sistent with the underlying ideas is that the linewidth of

the single-particle spectra of the cuprates abruptly ac-

quires a large elastic part in going from the overdoped

cuprates[19]to the underdoped cuprates.

The single-particle states at T→0in the stable TRS

breaking state in the absence of impurity scattering have

been derived[14,15]to have energies given by

=?k±D(k),for E k?μ,(1)

where D(k)≈D0(1?T/T g)1/2cos2(2φ)/[1+(?k/?c)2]is

the gap function,which is anisotropic and temperature-

dependent.φis the angle of?k and?k is the“band-

structure”energy of a tight-binding model to?t the

Fermi surface[20]found by ARPES for x≤x c,with

e?ective nearest-neighbor and next-nearest-neighbor Cu-

Cu hopping parameters t and t′respectively:

?k=?2t[cos k x+cos k y]?4t′cos k x cos k y,(2)

where t′/t=?0.35.?c is a cuto?of order D0.The simple

theory[15]gives D0≈√

πImΣ(k,ω)

√

ω2+π2T2.(6)

λis the dimensionless coupling constant used to?t

ARPES data for x≥x c by the MFL spectral function.

Near the zeroes of D(?k)(nodal region),one can again

use momentum and energy conservation to calculate the

phase space for decay.One easily?nds that the self-

energy forω?D(?k)is

?ImΣin(k,ω,T)=λ(ω2+π2T2)/D0,(7)

while forω?D0,it reverts to the MFL formτ?1

.An-

other interpolation form,such as in Eq.(5)is used to

connect the nodal region and the antinodal region.In

our calculations below,we have used Eq.(7)in an angu-

lar region extending toδφ=π/20from the nodal point

and Eq.(5)elsewhere.Similar results are obtained for

factors of2variations about this partitioning.For the

low energy region of interest,ReΣproduces negligible

corrections and has been ignored.

The spectral function is calculated using Eqs.(3,4).

The parameters used in the evaluation are D0/T g=2.5,

andλis?xed by the value determined by the MFL?ts

to the spectral function for x≥x c to be≈0.5.Earlier,

the speci?c heat and the magnetic susceptibility in the

underdoped cuprates were?tted[15]to experiments with

D0/T g≈2.5.Thus there are no free parameters left to

?t.

In Fig.1,the calculated spectral function is plotted

at the anti-nodal point for various T/T g and for various

angles at T/T g=0.5.The inset shows the de?nition

of various quantities used to represent the experiments

in Ref.[1].?(φ)is de?ned as the energy at which the

spectral function peaks below the chemical potential at

the angleφ,while I(0,φ)and I(?,φ)are the intensities

at the chemical potential(set toω=0)and at?,re-

spectively.Either?(φ)=0or1?I(0,φ)/I(?,φ)=0

-4

-2

A (k )

/D 0

-4

-2

FIG.1:The spectral functions (EDCs)for (a)the antinodal

angle at various temperatures;(b)?xed temperature (T /T g =0.5)while varying angles from antinode to node.Here,D 0=2.5T g .As shown in inset,we have followed the representation of the data as in experiments,explained in the main text.

implies that A (k F ,ω)peaks at the chemical potential,

giving the impression of ”Fermi surface”.The angular re-gions where these quantities vanish are de?ned as “Fermi arcs”.

The experimental results and the calculated values of

?(φ)/?(0)and of 1?I (0,φ)/I (?,φ)are shown in Fig.2.The central aspect of the experimental results deduced from the experiments,is shown in Fig.3together with the results of calculations.Fig.3shows that the Fermi arc length Φ(T/T g (x ))for a whole range of underdoped cuprates at various dopings x is a universal function of T/T g (x )and that at T →0,Φ→0.The theory in the pure limit is in quantitative accord with the experi-mental curve without any free parameters except for the region of T/T g ≈1.This disagreement may be traced to the fact that the experimental gap rises below T g much faster than the mean-?eld theory.The qualitatively new physics in the results deduced from the experiments is however in the low temperature limit.

The agreement actually is a little worse if we include impurity scattering,see also Fig. 3.We have calcu-lated the spectral function including the e?ects of small angle scattering due to impurities between the Cu-O planes [21,22].Due to the anisotropy of the density of states in the pseudogap state,the impurity scattering rate becomes anisotropic as well as frequency-dependent.With the impurity scattering estimated from experiments which gives an elastic scattering rate of about 200K at the nodal point above T g [19],we have to use D 0/T g ≈3to get agreement with the experiments.With small an-gle scattering,only four Fermi points remain at T →0but a weak smooth bump develops for these values at around T/T g ≈0.1,which however continues to give a theoretical curve within the experimental error bars.The impurity scattering,of course,varies also from sample to

010203040

(

)(0)

(deg.)

1-I (0)/I ,

)

(deg.)

FIG.2:The gap function and the peak intensity 1?I (0,φ)/I (?,φ)as functions of the angle.The solid lines are theoretical results for T /T g =0.1to 0.8(in sequence from right to left).

sample.We should also mention that if the large angle impurity scattering in the unitary limit were important,say due to impurities in the plane,we expect an impurity resonance at the nodal points,so that the Fermi points are smeared out.

We are not aware of any other theory giving the re-sults of the experiments,speci?cally a scaling form for the Fermi arc Φ(T/T g (x ))with Φ(0)→0.There exist theories which generate anisotropic reduction of the den-sity of states in the underdoped state without generating Fermi points at T =0,such as approximate solutions of the Hubbard model with dynamical mean-?eld theory and its extensions [23].There is no indication that the Fermi surface shrinks to four Fermi points in such cal-culations at T →0.This is consistent with the general arguments given in this paper that this is not possible without some symmetry breaking.On the other hand,experiments must be done in materials where the extrap-olation to the ground state of the pseudogap phase can be done more precisely.This necessitates experiments in high quality underdoped samples just at the boundary to superconductivity.Another prediction of the theory which awaits more precise measurements through higher energy and momentum resolution is that the angular de-pendence of the pseudogap is ∝(φ?φ0)2near the points φ0at which it is 0,[see the expression for the pseudogap following Eq.(1)].This is important to conclude that

0.00.5 1.0

(T /T

)/(2)

T/T

FIG.3:The Fermi arc length as a function of T /T g .The

experimental data from Ref.[1]are shown in red dots with error bars.The solid lines are theoretical results:one in the pure limit with D 0=2.5T g ;another with small angle impu-rity scattering,?tted with D 0=3T g and 1/τ0=T g ,where 1/τ0is the impurity scattering rate at the nodal point,and its magnitude can be estimated from the normal state data,such as in Ref.[19].

there is no phase associated with the pseudogap,unlike superconducting gaps or gaps due to change in transla-tional symmetry due to charge or spin-density waves or staggered ?ux phases.

In conclusion,we would like to re-emphasize that the scaling of the Fermi arc is a new phenomena in con-densed matter physics and that it requires a broken sym-metry.We have presented the results of a theory,one of whose consequences is a Fermi surface with only four points.With two parameters taken to ?t thermodynamic measurements,we have been able to account for the de-tails of the Fermi arcs.The same theory has predicted a time-reversal violating state in the pseudogap region and the theory of quantum critical ?uctuations [17]which gives the phenomenological spectrum [7]with which the “strange metal”phase is understood.These ?uctuations also give rise to attractive paring interaction in the “d-wave”symmetry [24].

Acknowledgement .One of us (CMV)is grateful to Leon Balents for a discussion of the Bloch’s counting the-orem.

[1]A.Kanigel et al.,Nature Physics 2,447(2006);

cond-mat/0605499.

[2]J.W.Loram et al.,J.Phys.Chem.Solids 62,59(2001);